7.2 Proposals for the holographic entanglement entropy of black holes

If one wants to generalize the proposal of Ryu and Takayanagi to black holes, the first step would be to find a (d + 1)-dimensional asymptotically-AdS metric, which solves the Einstein equations in bulk and whose conformal boundary describes a d-dimensional black hole. To find such a metric explicitly may be a difficult task, although some exact solutions are known. First of all, it is easy to construct an AdS metric, which gives a de Sitter spacetime on the boundary. The de Sitter horizon is in many aspects, similar to a black hole horizon. Entanglement entropy associated to the de Sitter horizon [130Jump To The Next Citation Point, 139Jump To The Next Citation Point] has the same properties as the entropy of any other Killing horizon. In four dimensions (d = 3) an exact solution to the Einstein equations has been found in [84Jump To The Next Citation Point] that describes a 3D black hole on the boundary. In three dimensions (d = 2), an exact solution, which describes a generic static two-dimensional black hole on the boundary, has been found in [194Jump To The Next Citation Point]. On the other hand, the results of [60Jump To The Next Citation Point] show that for any chosen metric on the boundary ∂M, one can find, at least in a small region close to the boundary, an exact solution to the Einstein equations with negative cosmological constant. Exact formulas are given in [60Jump To The Next Citation Point]. Thus, at least principally, it is not a problem to find an asymptotically AdS metric which describes a black hole on the boundary.

The next question is how to choose the minimal surface γ. Emparan’s proposal [83Jump To The Next Citation Point] consists of the following. Suppose the metric on the boundary of asymptotically-AdS spacetime describes a black hole with a Killing horizon at surface Σ. Presumably, the horizon on the boundary ∂M is extended to the bulk. The bulk horizon is characterized by vanishing extrinsic curvature and is a minimal (d − 1)-dimensional surface. Thus, one can choose the bulk horizon to be that minimal surface γ, the area of which should appear in the holographic formula (303View Equation). In this construction the Killing horizon Σ is the only boundary of the minimal surface γ. This prescription is perfectly eligible if one computes the entanglement entropy of a black hole in infinite spacetime. In [83Jump To The Next Citation Point] it was applied to the entropy of a black hole residing on the 2-brane in the 4d solution of [84Jump To The Next Citation Point]. In [139Jump To The Next Citation Point] a similar prescription is used to compute entanglement entropy of the de Sitter horizon.

On the other hand, in certain situations it is interesting to consider a black hole residing inside a cavity, the “black hole in a box”. Then, as we have learned in the two-dimensional case, the entanglement entropy will depend on the size L of the box so that, in the limit of large L, the entropy will have a thermal contribution proportional to the volume of the box. This contribution is additional to the pure entanglement part, which is due to the presence of the horizon Σ. In order to reproduce this dependence on the size of the box, one should use a different proposal. A relevant proposal was suggested by Solodukhin in [202Jump To The Next Citation Point].

Let a d-dimensional spherically-symmetric static black hole with horizon Σ lie on the regularized boundary (with regularization parameter 𝜖) of asymptotically anti-de Sitter spacetime AdSd+1 inside a spherical cavity ΣL of radius L. Consider a minimal d-surface Γ, whose boundary is the union of Σ and Σ L. Γ has saddle points, where the radial AdS coordinate has an extremum. By spherical symmetry the saddle points form a (d − 2)-surface 𝒞 with the geometry of a sphere. Consider the subset Γ h of Γ whose boundary is the union of Σ and 𝒞. According to the prescription of [202Jump To The Next Citation Point], the quantity

S = Area-(Γ-h) (304 ) 4Gd+N1
is equal to the entanglement entropy of the black hole in the boundary of AdS. In particular, it gives the expected dependence of the entropy on the UV regulator 𝜖. The minimal surface Γ h “knows” about the existence of the other boundary ΣL. That is why Eq. (304View Equation) reproduces correctly the dependence of the entropy on the size of the “box”. In [202Jump To The Next Citation Point] this proposal has been verified for d = 2 and d = 4.

It should be noted that as far as the UV divergent part of the entanglement entropy is concerned, the two proposals [83Jump To The Next Citation Point, 139] and [202Jump To The Next Citation Point] give the same result. This is due to the fact that the UV divergences come from that part of the minimal surface, which approaches the boundary ∂M of the AdS spacetime. In both proposals this part of the surfaces γ and Γ h is the same. Thus, the difference is in the finite terms, which are due to global properties of the minimal surface.

From the geometrical point of view, the holographic calculation of the logarithmic term in the entanglement entropy is related to the surface anomalies studied by Graham and Witten [123] (this point is discussed in [191]).

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